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%!TEX root = forallxyyc.tex
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Tabelas de verdade
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Tabelas de verdade características
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name "s:CharacteristicTruthTables"

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Qualquer sentença de LVF é composta de letras sentenciais, possivelmente
 combinadas usando-se conectivos sentenciais.
 O valor de verdade da sentença composta depende apenas do valor de verdade
 das letras sentenciais que a compõem.
 A fim de saber o valor de verdade de `
\begin_inset Formula $(D\eand E)$
\end_inset

',, por exemplo, voê precisa somente saber o valor de verdade de `
\begin_inset Formula $D$
\end_inset

' e o valor de verdade de `
\begin_inset Formula $E$
\end_inset

'.
 Introduzimos cinco conectivos no capítulo 
\begin_inset CommandInset ref
LatexCommand ref
reference "s:TFLConnectives"
plural "false"
caps "false"
noprefix "false"

\end_inset

, assim precisamos simplesmente explicar como eles mapeiam entre valores
 de verdade.
 Por conveniência, abreviaremos `Verdadeiro' por `V' e `Falso' por `F' (mas
 para ser claro, os dois valores de verdade são o Verdadeiro e o Falso;
 os valores de verdade não são 
\emph on
letras
\emph default
).
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A
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: se 
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meta
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{
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A
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}
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 for verdadeira, então 
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{
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A
\begin_inset ERT
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}
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 será falsa.
 Se 
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{
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A
\begin_inset ERT
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}
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 for verdadeira, então 
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meta
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\begin_inset ERT
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{
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A
\begin_inset ERT
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}
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 será falsa.
 Podemos resumir isto na 
\emph on
seguinte tabela de verdade característica
\emph default
 para negação: 
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A
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{
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A
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V 
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F 
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V 
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A
\begin_inset ERT
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}
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 e 
\begin_inset ERT
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\backslash
meta
\end_layout

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\begin_inset ERT
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{
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B
\begin_inset ERT
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}
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, 
\begin_inset ERT
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\backslash
meta
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{
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A
\begin_inset ERT
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eand
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meta
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{
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B
\begin_inset ERT
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}
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 será verdadeira se e somente se tanto 
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\begin_inset ERT
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{
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A
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 como 
\begin_inset ERT
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meta
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\begin_inset ERT
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{
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B
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}
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 forem verdadeiras.
 Podemos resumir isto na 
\emph on
seguinte tabela de verdade característica
\emph default
 para conjunção: 
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A
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{
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B
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V 
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V 
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\begin_inset Text

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V
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<row>
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V 
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F 
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F
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F 
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\begin_inset Text

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V 
\end_layout

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<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

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F
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<row>
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F 
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F 
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F 
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\begin_layout Standard
Note que a conjunção é 
\emph on
simétrica
\emph default
.
 O valor de verdade de 
\begin_inset Formula $\meta{A}\eand\meta{B}$
\end_inset

 é sempre o mesmo que o valor de verdade de 
\begin_inset Formula $\meta{B}\eand\meta{A}$
\end_inset

.
 
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Disjunção
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\begin_layout Standard
Lembre-se de que `
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' sempre representa a disjunção inclusivo.
 Desse modo, para quaisquer sentença 
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\backslash
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{
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A
\begin_inset ERT
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}
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 e 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

, 
\begin_inset Formula $\meta{A}\eor\meta{B}$
\end_inset

 será verdadeira se e somente se ou 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
meta
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\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
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A
\begin_inset ERT
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}
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 ou 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 forem verdadeiras.
 Podemos resumir isto na 
\emph on
seguinte tabela de verdade característica
\emph default
 para disjunção: 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
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<row>
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\backslash
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{
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A
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\begin_inset ERT
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{
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B
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V 
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V 
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V
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V 
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F 
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<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
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\end_inset
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<row>
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\begin_inset Text

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F 
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
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V 
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V
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F 
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F 
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\begin_layout Standard
Como a conjunção, a disjunção é simétrica.
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\begin_layout Paragraph
Condicional
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\begin_layout Standard
Deixando bastante claro, admitimos: condicionais são um antigo problema
 em LVF.
 O quanto os condicionais são problemáticos é filosoficamente controverso.
 Discutiremos algumas sutilezas em 
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\begin_inset CommandInset ref
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reference "s:IndicativeSubjunctive"
plural "false"
caps "false"
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 e 
\begin_inset CommandInset ref
LatexCommand ref
reference "s:ParadoxesOfMaterialConditional"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Por enquanto, iremos estipular o seguinte: 
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset

 será falsa se e somente se 
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\begin_layout Plain Layout


\backslash
meta
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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A
\begin_inset ERT
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}
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 for verdadeira e 
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\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 for falsa.
 Podemos resumir isto com 
\emph on
seguinte tabela de verdade característica
\emph default
 para o condicional.
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
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\begin_inset ERT
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\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

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{
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B
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</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
O condicional é 
\emph on
assimétrico
\emph default
.
 Você não pode trocar o antecedente e consequente sem mudar o significado
 da sentença, pois 
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset

 tem uma tabela de verdade muito diferente de 
\begin_inset Formula $\meta{B}\eif\meta{A}$
\end_inset

.
 
\end_layout

\begin_layout Paragraph
Bicondicional
\end_layout

\begin_layout Standard
Uma vez que o bicondicional é o mesmo que a conjunção de um condicional
 em ambas as direções, desejamos que a tabela de verdade para o bicondicional
 seja: 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\meta{A}\eiff\meta{B}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
Sem surpresas, o bicondicional é simétrico.
\end_layout

\begin_layout Chapter
Conectivos verofucionais
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:TruthFunctionality"

\end_inset


\end_layout

\begin_layout Section
A ideia de verofuncionalidade
\end_layout

\begin_layout Standard
Vamos introduzir uma importante ideia.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 Um conectivo é 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

verofuncional
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 sse o valor de verdade de uma sentença com este conectivo como operador
 principal é unicamente determinado pelo(s) valore(s) de verdade da(s) sentença(
s) constituinte(s).
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
Todo conectivo em LVF é verofuncional.
 O valor de verdade de uma negação é unicamente determinada pelo valor de
 verdade da sentença que não é negada.
 O valor de verdade de uma conjunção é unicamente determinado pelo valor
 de verdade de ambos conjunctos.
 O valor de verdade da disjunção é unicamente determinado pelo valor de
 verdade de amos disjuntos e assim por diante.
 Para determinar o valor de verdade de alguma sentença de LVF, precisamos
 apenas saber o valor de verdade dos componentes dela.
\end_layout

\begin_layout Standard
É isto que dá à LVF seu nome: ela é 
\emph on
verofuncional
\emph default
.
\end_layout

\begin_layout Standard
Em muitas linguagens, há conectivos que não são verofuncionais.
 No Português, por exemplo, podemos formar uma nova sentença a partir de
 sentença mais simples, colocando na frente dela `é necessariamente o caso
 que\SpecialChar ldots
'.
 O valor de verdade desta nova sentença não é fixada apenas pelo valor de
 verdade da sentença original.
 Considere, pois, duas sentenças verdadeiras: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $2+2=4$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Shostakovich escreveu quinze quartetos de cordas
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Enquanto é necessariamente o caso que 
\begin_inset Formula $2+2=4$
\end_inset

, não é 
\emph on
necessariamente
\emph default
 o caso que Shostakovich escreveu quinze quartetos de cordas.
 Se Shostakovich tivesse morrido prematuramente, ele não teria terminado
 o Quarteto número 15; se ele tivesse vivido mais, poderia ter escrito alguns
 outros quartetos.
 Destarte, `é necessariamente o caso que\SpecialChar ldots
' é um conectivo do Português, mas
 não é 
\emph on
verofuncional
\emph default
.
\end_layout

\begin_layout Section
Simbolizar versus traduzir 
\end_layout

\begin_layout Standard
Todos os conectivos de LVF são verofuncionais, mas mais do que isso: de
 fato, eles não fazem nada, exceto fazer um mape entre valores de verdade.
 
\end_layout

\begin_layout Standard
Quando simbolizamos uma sentença ou argumento em LVF, ignoramos tudo, exceto
 a contribuição que os valores de verdade de um componente poderiam dar
 para calcular o valor de verdade da sentença toda.
 Existem sutilezas em nossas reivindicações cotidianas que estão além dos
 meros valores de verdade: sarcasmo; poesia; malícia [
\emph on
snide implicature
\emph default
]; ênfase.
 Estas são partes importantes do discurso cotidiano, mas nada disso é mantido
 em LVF.
\end_layout

\begin_layout Standard
Como foi observado em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:TFLConnectives"
plural "false"
caps "false"
noprefix "false"

\end_inset

, LVF não pode capturar as diferenças sutis entre as seguintes sentenças
 do Português: 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Dana é um lógico e Dana é uma pessoa legal.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Embora Dana seja lógico, Dana é uma pessoa legal.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Dana é um lógico, apesar de ser uma pessoa legal.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Dana é uma pessoa lega, mas também é lógico.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Não obstante o fato de Dana ser lógico, ele é uma pessoa legal.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Todas as sentença acima serão simbolizadas pela mesma sentença de LVF,
 talvez `
\begin_inset Formula $L\eand N$
\end_inset

'.
 Continuaremos dizendo que usamos sentenças de LVF para 
\emph on
simbolizar
\emph default
 sentenças do Português.
 Muitos outros manuais falam sobre 
\emph on
traduzir
\emph default
 sentenças do Português para LVF.
 Todavia, uma boa tradução deveria preservar certas facetas de significado
 e — como acabamos de apontar — LVF não pode mesmo fazer isso.Por causa disso,
 falaremos de 
\emph on
simbolizar
\emph default
 sentenças do Português, em vez de 
\emph on
traduzi
\emph default
-las.
 Isto afeta como deveríamos entender nossa chave de simbolização.
 Considere uma chave como: 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{ekey}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
item[L]
\end_layout

\end_inset

 Dana é um lógico.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
item[N]
\end_layout

\end_inset

 Dana é uma pessoa legal.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{ekey}
\end_layout

\end_inset

 Outros manuais entenderão isto como uma estipulação de que a sentença de
 LVF `
\begin_inset Formula $L$
\end_inset

' deveria 
\emph on
significar
\emph default
 que Dana é um lógico e que a sentença de LVF `
\begin_inset Formula $N$
\end_inset

' deveria 
\emph on
significar
\emph default
 que Dana é uma pessoa legal.
 Contudo, LVF não é, de forma alguma, equipada para lidar com 
\emph on
significado
\emph default
.
 A chave de simbolização precedente está fazendo nem mais nem menos do que
 estipular que a sentença de LVF `
\begin_inset Formula $L$
\end_inset

' deveria tomar o mesmo valor de verdade que a sentença do Português `Dana
 é um lógico' (seja qual valor poderia ser) e que a sentença de LVF `
\begin_inset Formula $N$
\end_inset

' deveria tomar o mesmo valor de verdade qie a sentença do Português `Dana
 é uma pessoa legal' (seja qual valor poderia ser).
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 Quando tratamos uma sentença de LVF como simbolizando uma sentença do Português
, estamos estipulando que a sentença de LVF tem de tomar o mesmo valor de
 verdade que o da sentença do Português.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Condicional indicativo versus condicional subjuntivo
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:IndicativeSubjunctive"

\end_inset

 Queremos retornar ao ponto segundo o qual LVF pode 
\emph on
somente
\emph default
 lidar com funções de verdade, considerando o caso do condicional.
 Quando introduzimos a tabela de verdade característica para o condicional
 materia em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:CharacteristicTruthTables"
plural "false"
caps "false"
noprefix "false"

\end_inset

, não dissemos qualquer coisa para justificá-la.
 Iremos oferecer agora uma justificação, que segue Dorothy Edginton
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Dorothy Edgington, `Conditionals', 2006, In: 
\emph on
Stanford Encyclopedia of Philosophy
\emph default
 (
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

http://plato.stanford.edu/entries/conditionals/
\end_layout

\end_inset

).
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
Suponha que Lara tenha desenhado algumas formas em um pedaço de papel e
 coloriu o interior de algumas delas.
 Nós não as vimos, mas, não obstante, reivindicamos: 
\end_layout

\begin_layout Quote
Se alguma forma é cinza, então esta forma é também círcular.
\end_layout

\begin_layout Standard
Acontece que Lara desenhou o seguinte:
\end_layout

\begin_layout Standard
\align center
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{tikzpicture}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

	
\backslash
node[circle, grey_shape] (cat1) {A};
\end_layout

\begin_layout Plain Layout

	
\backslash
node[right=10pt of cat1, diamond, phantom_shape] (cat2)  { } ;
\end_layout

\begin_layout Plain Layout

	
\backslash
node[right=10pt of cat2, circle, white_shape] (cat3)  {C} ;
\end_layout

\begin_layout Plain Layout

	
\backslash
node[right=10pt of cat3, diamond, white_shape] (cat4)  {D};
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{tikzpicture}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
Neste caso, nossa reivindicação é certamente verdadeira.
 As formas C e D não são cinzas e dificilmente podem ser apresentadas como
 
\emph on
contraexemplos
\emph default
 a nossa reivindicação.
 A forma A 
\emph on
é
\emph default
 cinza, mas, felizmente, ela também é circular.
 Desse modo, nossa reivindicação não tem contraexemplos.
 Ela deve ser verdadeira.
 Isto significa que cada uma das seguintes 
\emph on
instâncias
\emph default
 de nossa reivindicação dever ser também verdadeiras: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{ebullet}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Se A é cinza, então A é circular (antecedente verdadeiro, consequente verdadeiro
).
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Se C é cinza, então C é circular (antecedente falso, consequente verdadeiro).
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Se D é cinza, então D é circular (antecedente falso, consequente falso).
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{ebullet}
\end_layout

\end_inset

 Entretanto, se Lara tivesse desenhado uma quarta forma, assim:
\end_layout

\begin_layout Standard
\align center
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{tikzpicture}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

	
\backslash
node[circle, grey_shape] (cat1) {A};
\end_layout

\begin_layout Plain Layout

	
\backslash
node[right=10pt of cat1, diamond, grey_shape] (cat2)  {B};
\end_layout

\begin_layout Plain Layout

	
\backslash
node[right=10pt of cat2, circle, white_shape] (cat3)  {C};
\end_layout

\begin_layout Plain Layout

	
\backslash
node[right=10pt of cat3, diamond, white_shape] (cat4)  {D};
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{tikzpicture}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\noindent
então nossa reivindicação seria falsa.
 Desse maneira, deve ser o caso que esta reivindicação é falsa:
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{ebullet}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Se B é cinza, então B é circular (antecedente verdadeiro, consequente falso).
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{ebullet}
\end_layout

\end_inset

 Ora, lembre-se de que todo conectivo de LVF tem de ser verofuncional.
 Isto significa meramente que os valores de verdade do antecedente e do
 consequente devem determinar unicamente o valor de verdade do condicional
 como um todo.
 Assim, a partir dos valores de verdade de nossas quatro reivindicações
 — que nos fornece com todas as combinações possíveis de verdade e falsidade
 no antecedente e no consequente —, podemos ler a tabela de verdade para
 o condicional material.
 
\end_layout

\begin_layout Standard
O que este argumento mostra é que `
\begin_inset Formula $\eif$
\end_inset

' é o 
\emph on
melhor
\emph default
 candidato para o condicional verofuncional.
 Dizendo de outra forma, 
\emph on
é o melhor condicional que LVF pode fornecer
\emph default
.
 Mas ele é bom como um substituto para os condicionais que usamos na linguagem
 cotidiana? Considere as duas sentenças: 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
item[
\backslash
ex{brownwins1}]
\end_layout

\end_inset

Se Mitt Romney tivesse vencido a eleição de 2012, ele teria sido o 45
\begin_inset Formula $^{o}$
\end_inset

 Presidente dos Estados Unidos.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
item[
\backslash
ex{brownwins2}]
\end_layout

\end_inset

Se Mitt Romney tivesse vencido a eleição de 2012, então ele teria se transformad
o em um balão a hélio e flutuado para o céu noturno.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 A sentença 
\begin_inset CommandInset ref
LatexCommand ref
reference "brownwins1"
plural "false"
caps "false"
noprefix "false"

\end_inset

 é verdadeira; a sentença 
\begin_inset CommandInset ref
LatexCommand ref
reference "brownwins2"
plural "false"
caps "false"
noprefix "false"

\end_inset

 é falsa, mas ambas têm os antecedentes falsos e os consequentes falsos.
 Assim, o valor de verdade da sentença inteira não é unicamente determinado
 pelo valor de verdade das partes.
 Não assuma livremente que você possa simbolizar adequadamente um `se…,então…'
 do Português por `
\begin_inset Formula $\eif$
\end_inset

' de LVF.
\end_layout

\begin_layout Standard
A questão crucial é que as sentenças 
\begin_inset CommandInset ref
LatexCommand ref
reference "brownwins1"
plural "false"
caps "false"
noprefix "false"

\end_inset

 e 
\begin_inset CommandInset ref
LatexCommand ref
reference "brownwins2"
plural "false"
caps "false"
noprefix "false"

\end_inset

 empregam condicionais 
\emph on
subjuntivos
\emph default
 em vez de condicionais 
\emph on
indicativos
\emph default
.
 Eles nos pedem para imaginar algo contrário ao fato — Mitt Romney perdeu
 a eleição de 2012 — e, então, pede-nos para avaliar que 
\emph on
teria
\emph default
 acontecido neste caso.
 Não é possível lidar com tais considerações, usando `
\begin_inset Formula $\eif$
\end_inset

'.
 
\end_layout

\begin_layout Standard
Diremos mais sobre as dificuldades com condicionais em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:ParadoxesOfMaterialConditional"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Por enquanto, contentar-nos-emos com a observação de que `
\begin_inset Formula $\eif$
\end_inset

' é o único candidato para o condicional verofuncional de LVF, mas que muitos
 condicionais do Português não podem ser adequadamente representados , usando-se
 `
\begin_inset Formula $\eif$
\end_inset

'.
 LVF é uma linguagem intrisicamente limitada.
 
\end_layout

\begin_layout Chapter
Tabelas de verdade completas
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:CompleteTruthTables"

\end_inset

 
\end_layout

\begin_layout Standard
Até agora, consideramos atribuir valores de verdade às sentenças de LVF
 indiretamente.
 Por exemplo, dissemos que uma sentença de LVF tal como `
\begin_inset Formula $B$
\end_inset

' tem de tomar o mesmo valor de verdade que o da sentença do Português `Big
 Ben está em Londres' (seja qual for o valor de verdade), mas também podemos
 atribuir valores de verdade 
\emph on
diretamente
\emph default
.
 Podemos simplesmente estipular que `
\begin_inset Formula $B$
\end_inset

' tem de ser verdadeira ou estipular que ela tem de ser falsa.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 Uma 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

valora
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
c{c}
\backslash
~a
\end_layout

\end_inset

o
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 é qualquer atribuição de valores de verdade a uma letra sentencial particular
 de LVF.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
O poder das tabelas de verdade encontra-se no seguinte.
 Cada linha de uma tabela de verdade representa uma valoração possível.
 A tabela de verdade inteira representa todas as valorações possíveis; desse
 modo, a tabela de verdade fornece-nos com um meio de calcular os valores
 de verdade de sentenças complexas em cada valoração possível.
 Isto é mais fácil de explicar com exemplos.
\end_layout

\begin_layout Section
Trabalhando com um exemplo [
\emph on
a worked example
\emph default
]
\end_layout

\begin_layout Standard
Considere a sentença `
\begin_inset Formula $(H\eand I)\eif H$
\end_inset

'.
 Há quatro maneiras possíveis de atribuir Verdadeiro e Falso às letras sentencia
is `
\begin_inset Formula $H$
\end_inset

' e `
\begin_inset Formula $I$
\end_inset

' — quatro valorações possíveis — que podemos representar com se segue:
 
\end_layout

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\align center
\begin_inset Tabular
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<column alignment="none" valignment="top" special="f">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
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\begin_layout Plain Layout
\begin_inset Formula $(H$
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\end_layout

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<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $I)$
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\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
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\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
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\end_layout

\end_inset


\end_layout

\end_inset
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<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H$
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\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
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\end_inset
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<cell alignment="none" valignment="top" topline="true" usebox="none">
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<cell alignment="none" valignment="top" topline="true" usebox="none">
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<row>
<cell alignment="center" valignment="top" usebox="none">
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
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F
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\end_inset
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<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

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</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

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\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

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\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

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\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
Para calcular o valor de verdade da sentença inteira `
\begin_inset Formula $(H\eand I)\eif H$
\end_inset

', copiamos, em primeiro lugar, os valores de verdade para as letras sentenciais
 e os escrevemos embaixo das letras na sentença.
 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $I$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(H$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
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\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $I)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
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}
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\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

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V
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}
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\end_layout

\end_inset
</cell>
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\begin_inset Text

\begin_layout Plain Layout

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\begin_layout Plain Layout
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{
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V
\begin_inset ERT
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}
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\end_layout

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</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

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V
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</cell>
<cell alignment="none" valignment="top" usebox="none">
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\begin_inset Text

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F
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</cell>
<cell alignment="none" valignment="top" usebox="none">
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</cell>
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<row>
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\begin_inset Text

\begin_layout Plain Layout
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\end_inset

 
\end_layout

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</cell>
<cell alignment="none" valignment="top" usebox="none">
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</cell>
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<row>
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</cell>
<cell alignment="none" valignment="top" usebox="none">
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\end_layout

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\begin_inset Text

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{
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F
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\end_inset

 
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\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

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<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

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\end_inset

 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
Considere agora a subsentença `
\begin_inset Formula $(H\eand I)$
\end_inset

'.
 Isto é uma conjunção 
\begin_inset Formula $(\meta{A}\eand\meta{B})$
\end_inset

, onde `
\begin_inset Formula $H$
\end_inset

' substitui 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e `
\begin_inset Formula $I$
\end_inset

' substitui 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 A tabela de verdade característica para a conjunção dá as condições de
 verdade para 
\emph on
qualquer
\emph default
 sentença da forma 
\begin_inset Formula $(\meta{A}\eand\meta{B})$
\end_inset

, independente do que poderiam ser 
\begin_inset Formula $\meta{A}$
\end_inset

 e 
\begin_inset Formula $\meta{B}$
\end_inset

.
 A tabela representa o ponto no qual uma conjunção é verdadeira sse ambos
 conjunctos são verdadeiros.
 Neste caso, nossos conjunctos são exatamente `
\begin_inset Formula $H$
\end_inset

' and `
\begin_inset Formula $I$
\end_inset

'.
 Eles são ambos verdadeiros na primeira linha da tabela de verdade (e somente
 nela).
 De acordo com isso, podemos calcular o valor de verdade da conjunção em
 todas as quatro linhas.
 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

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\backslash
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\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $I$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(H$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $I)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
Agora, a sentença inteira com a qual estamos lidando é um condicional 
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset

, onde `
\begin_inset Formula $(H\eand I)$
\end_inset

' substitui 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e `
\begin_inset Formula $H$
\end_inset

' substitui 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 Na segunda linha, por exemplo, `
\begin_inset Formula $(H\eand I)$
\end_inset

' é falsa e `
\begin_inset Formula $H$
\end_inset

' é verdadeira.
 Uma vez que o condicional é verdadeiro quando o antecedente é falso, escrevemos
 um `V' na segunda linha embaixo do símbolo do condicional.
 Continuando para as outras três linhas, obtemos: 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $I$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(H$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $I)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
O condicional é o operador lógico principal da sentença, assim a coluna
 de `V's embaixo do condicional nos diz que a sentença `
\begin_inset Formula $(H\eand I)\eif H$
\end_inset

' é verdadeira, independentemente dos valores de verdade de `
\begin_inset Formula $H$
\end_inset

' e `
\begin_inset Formula $I$
\end_inset

'.
 Elas podem ser verdadeiras ou falsas em qualquer combinação e a sentença
 composta ainda continuará verdadeira.
 Uma vez que consideramos todas as quatro possíveis atrinuições de verdade
 e falsidade a `
\begin_inset Formula $H$
\end_inset

' e `
\begin_inset Formula $I$
\end_inset

' — ou seja, uma vez que consideramos todas as diferentes valorações —,
 podemos dizer que `
\begin_inset Formula $(H\eand I)\eif H$
\end_inset

' é verdadeira em qualquer valoração.
 Neste exemplo, não repetimos todas as entradas em qualquer coluna em qualquer
 tabela sucessiva.
 Na realidade, entretanto, quando escrevemos tabelas de verdade no papel,
 não é prático apagar colunas inteiras ou reescrever a tabela inteira a
 cada passo.
 Embora fique mais volumosa, a tabela de verdade pode ser escrita da seguinte
 maneira: 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $I$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(H$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $I)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
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Muitas das colunas embaixo da sentença estão lá apenas para propósitos de
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\emph on
for bookkeeping purposes
\emph default
].
 A coluna que mais importa é a coluna debaixo do 
\emph on
operador lógico principal
\emph default
 da sentença, uma vez que isto lhe diz o valor de verdade da sentença inteira.
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\begin_layout Section
Construindo tabelas de verdade completas
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Uma 
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\backslash
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tabela de verdade completa
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\emph on
valoração
\emph default
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s.
\end_layout

\begin_layout Standard
O tamanho da tabela de verdade completa depende do número das diferentes
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 Isto é verdadeiro mesmo se a mesma letra é repetida várias vezes, como
 na sentença `
\begin_inset Formula $[(C\eiff C)\eif C]\eand\enot(C\eif C)$
\end_inset

'.
 A tabela de verdade completa exige somente duas linhas, porque há apenas
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\begin_inset Formula $C$
\end_inset

' pode ser verdadeira ou pode ser falsa.
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<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
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\end_layout

\begin_layout Standard
Olhando para coluna embaixo do operador lógico principal, vemos que a sentença
 é falsa em ambas as linhas da tabela; ou seja, a sentença é falsa, independemen
te de se `
\begin_inset Formula $C$
\end_inset

' é verdadeiro ou falso.
 Ela é falsa em qualquer valoração.
 A sentença que contém duas letras sentenciais exige quatro linhas para
 uma tabela de verdade completa como nas tabelas de verdade características
 para os quatro conectivos binários e como na tabela de verdade completa
 para `
\begin_inset Formula $(H\eand I)\eif H$
\end_inset

'.
 
\end_layout

\begin_layout Standard
Uma sentença que contém três letras sentenciais exige oito linhas 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="9" columns="8">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<row>
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\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
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<cell alignment="none" valignment="top" usebox="none">
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\begin_layout Plain Layout
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\begin_inset ERT
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\backslash
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\begin_layout Plain Layout
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V 
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\end_layout

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</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
A partir desta tabela, sabemos que a sentença `
\begin_inset Formula $M\eand(N\eor P)$
\end_inset

' pode ser verdadeira ou falsa, dependendo dos valores de verdade de `
\begin_inset Formula $M$
\end_inset

', `
\begin_inset Formula $N$
\end_inset

' e `
\begin_inset Formula $P$
\end_inset

'.
 Uma tabela de verdade completa para uma sentença que contém quatro diferentes
 letras sentenciais exige 16 linhas.
 Cinco letras, 32 linhas.
 Seis letras, 64 linhas e assim por diante.
 Sendo completamente geral: se uma tabela de verdade tem 
\begin_inset Formula $n$
\end_inset

 diferentes letras sentenciais, então devem existir 
\begin_inset Formula $2^{n}$
\end_inset

 linhas.
 
\end_layout

\begin_layout Standard
Com o intuito de preencher as colunas de uma tabela de verdade completa,
 comece com a letra sentencial mais à direita e alterne `V' e `F'.
 Na próxima coluna à esquerda, escreve dois `V's e escreva escreva dois
 `F's e repita o procedimento.
 Para a terceira letra sentencial, escreva quatro `V's seguidos por quatro
 `F's.
 Isto produz uma tabela de verdade com oito linhas como a de cima.
 Para uma tabela com 16 linhas, a coluna a seguir de letras sentenciais
 deveriam ter oitos `V's seguidos por oito `F's.
 Para uma tabela com 32 linhas, a coluna a seguir teria 16 `V's seguidos
 por 16 `F's e assim por diante.
 
\end_layout

\begin_layout Section
Mais sobre parênteses
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:MoreBracketingConventions"

\end_inset

 Considere estas duas sentenças:
\begin_inset Formula 
\begin{align*}
((A\eand B)\eand C)\\
(A\eand(B\eand C))
\end{align*}

\end_inset

Estas são verofuncionalmente equivalentes.
 Consequentemente, nunca fará qualquer diferença da perspectiva do valor
 de verdade — que é tudo que importa para LVF (veja 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:TruthFunctionality"
plural "false"
caps "false"
noprefix "false"

\end_inset

) — qual das duas sentenças afirmamos (ou negamos).
 Ainda que a ordem dos parênteses não importa no que diz respeito à verdade
 delas.
 não deveríamos excluí-los.
 A expressão 
\begin_inset Formula 
\begin{align*}
A\eand B\eand C
\end{align*}

\end_inset

é ambígua entre as duas sentenças acima.
 A mesma observação vale para disjunções.
 As seguintes sentenças são logicamente equivalentes: 
\begin_inset Formula 
\begin{align*}
((A\eor B)\eor C)\\
(A\eor(B\eor C))
\end{align*}

\end_inset

Mas não deveríamos simplesmente escrever:
\begin_inset Formula 
\begin{align*}
A\eor B\eor C
\end{align*}

\end_inset

Na realidade, isso é um fato específico sobre a tabela de verdade característica
 de 
\begin_inset Formula $\eor$
\end_inset

 e 
\begin_inset Formula $\eand$
\end_inset

 que garante que quaisquer duas conjunções (ou disjunções) contendo as mesmas
 sentenças são verofuncionalmente equivalentes, independentemente de como
 você coloca os parênteses.
 
\emph on
Isto é apenas verdadeiro de conjunções e disjunções
\emph default
, entretanto.
 As seguintes duas sentenças têm 
\emph on
diferentes
\emph default
 tabelas de verdade: 
\begin_inset Formula 
\begin{align*}
((A\eif B)\eif C)\\
(A\eif(B\eif C))
\end{align*}

\end_inset

Desse modo, se escrevêssemos:
\begin_inset Formula 
\begin{align*}
A\eif B\eif C
\end{align*}

\end_inset

ela seria perigosamente ambígua.
 Excluir os parênteses seria desastroso.
 Similarmente, estas sentenças têm diferentes tabelas de verdade: 
\begin_inset Formula 
\begin{align*}
((A\eor B)\eand C)\\
(A\eor(B\eand C))
\end{align*}

\end_inset

Assim, se escrevêssemos:
\begin_inset Formula 
\begin{align*}
A\eor B\eand C
\end{align*}

\end_inset

isso seria perigosamente ambíguo.
 
\emph on
Nunca escreva isto
\emph default
.
 A moral é: nunca exclua os parênteses (exceto os mais externos).
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
practiceproblems
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "pr.TT.TTorC"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Apresente as tabelas de verdade completas para cada uma das seguintes sentenças
:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $C\eif\enot C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eiff B)\eiff\enot(A\eiff\enot B)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eif B)\eor(B\eif A)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eand B)\eif(B\eor A)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eor B)\eiff(\enot A\eand\enot B)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\bigl[(A\eand B)\eand\enot(A\eand B)\bigr]\eand C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $[(A\eand B)\eand C]\eif B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot\bigl[(C\eor A)\eor B\bigr]$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Cheque todas as reivindicações que foram feitas na introdução das novas
 convenções de notação em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:MoreBracketingConventions"
plural "false"
caps "false"
noprefix "false"

\end_inset

, ou seja, mostre que:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

`
\begin_inset Formula $((A\eand B)\eand C)$
\end_inset

' and `
\begin_inset Formula $(A\eand(B\eand C))$
\end_inset

' têm a mesma tabela de verdade
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

`
\begin_inset Formula $((A\eor B)\eor C)$
\end_inset

' and `
\begin_inset Formula $(A\eor(B\eor C))$
\end_inset

' têm a mesma tabela de verdade
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

`
\begin_inset Formula $((A\eor B)\eand C)$
\end_inset

' and `
\begin_inset Formula $(A\eor(B\eand C))$
\end_inset

' não têm a mesma tabela de verdade
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

`
\begin_inset Formula $((A\eif B)\eif C)$
\end_inset

' and `
\begin_inset Formula $(A\eif(B\eif C))$
\end_inset

' não têm a mesma tabela de verdade
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Cheque também se:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
item[5.]
\end_layout

\end_inset

 `
\begin_inset Formula $((A\eiff B)\eiff C)$
\end_inset

' and `
\begin_inset Formula $(A\eiff(B\eiff C))$
\end_inset

' têm a mesma tabela de verdade
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Escreva as tabelas de verdade completas das seguintes sentenças e marque
 a coluna que representa os valores de verdade possíveis da sentença inteira.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
par 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(S\eiff(P\eif S))$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
par 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot[(X\eand Y)\eor(X\eor Y)]$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
par 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eif B)\eiff(\enot B\eiff\enot A)$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
par 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $[C\eiff(D\eor E)]\eand\enot C$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
par 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(G\eand(B\eand H))\eiff(G\eor(B\eor H))$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Escreva as tabelas de verdade completas das seguintes sentenças e marque
 a coluna que representa os valores de verdade possíveis da sentença inteira.
 
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begin{earg}
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\begin_layout Standard
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item 
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\begin_inset Formula $(D\eand\enot D)\eif G$
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par 
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\begin_layout Standard
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\end_layout

\begin_layout Chapter
Conceitos semânticos
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LatexCommand label
name "s:SemanticConcepts"

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\end_layout

\begin_layout Standard
Nos capítulos anteriores, introduzimos a ideia de valoração e mostramos
 como determinar o valor de verdade de qualquer sentença de LVF em qualquer
 valoração, usando a tabela de verdade.
 Neste capítulo, introduziremos algumas ideias relacionadas e mostraremos
 como usar as tabelas de verdade para testar se estas ideias se aplicam
 ou não.
\end_layout

\begin_layout Section
Tautologias e contradições
\end_layout

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Em 
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S
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\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicNotions"
plural "false"
caps "false"
noprefix "false"

\end_inset

, explicamos 
\emph on
verdade necessária
\emph default
 e 
\emph on
falsidade necessária
\emph default
.
 Ambas noções têm representantes em LVF.
 Começaremos com um representante para verdade necessária.
 
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{
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 é uma 
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status collapsed

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\backslash
define
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\end_inset


\begin_inset ERT
status collapsed

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{
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\end_inset

tautologia
\begin_inset ERT
status collapsed

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}
\end_layout

\end_inset

 (em LVF) sse ela é verdadeira em qualquer valoração.
 
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status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\end_layout

\begin_layout Standard
Podemos determinar se uma sentença é uma tautologia, usando apenas tabelas
 de verdade.
 Se a sentença é verdadeira em qualquer linha de uma tabela de verdade completa,
 então ela é verdadeira em qualquer valoração e, desse modo, ela é uma tautologi
a.
 No exemplo de 
\begin_inset ERT
status collapsed

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\backslash
S
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\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:CompleteTruthTables"
plural "false"
caps "false"
noprefix "false"

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, the sentence is true on every line of a complete truth table, then it
 is true on every valuation, so it is a tautology.
 In the example of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
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\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:CompleteTruthTables"
plural "false"
caps "false"
noprefix "false"

\end_inset

, `
\begin_inset Formula $(H\eand I)\eif H$
\end_inset

' é uma tautologia.
 Entretanto, isto é somente um representante para verdade necessária.
 Há algumas verdades necessárias que não podem ser adequadamente simbolizadas
 em LVF.
 Um exemplo é `
\begin_inset Formula $2+2=4$
\end_inset

'.
 Isto 
\emph on
deve
\emph default
 ser verdadeiro, mas se tentarmos simbolizá-lo em LVF, o melhor que podemos
 oferecer é uma letra sentencial e nenhuma letra sentencial é uma tautologia.
 Ainda assim, se pudermos adequadamente simbolizar algumas sentenças do
 Português usando sentenças de LVF que é uma tautologia, então esta sentença
 do Português expressa uma verdade necessária.
\end_layout

\begin_layout Standard
Temos um representante similar para falsidade necessária: 
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status collapsed

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\backslash
factoidbox
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset

 
\begin_inset Formula $\meta{A}$
\end_inset

 é uma 
\begin_inset ERT
status collapsed

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\backslash
define
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset

contradi
\begin_inset ERT
status open

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\backslash
c{c}
\backslash
~a
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\end_inset

o
\begin_inset ERT
status collapsed

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}
\end_layout

\end_inset

 (em LVF) sse ela é falsa em qualquer valoração.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Podemos determinar se uma sentença é uma contradição usando apenas tabelas
 de verdade.
 Se uma sentença é falsa em qualquer linha de uma tabela de verdade completa,
 então ela é falsa em qualquer valoração e, desse modo, ela é uma contradição.
 No exemplo de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:CompleteTruthTables"
plural "false"
caps "false"
noprefix "false"

\end_inset

, `
\begin_inset Formula $[(C\eiff C)\eif C]\eand\enot(C\eif C)$
\end_inset

' é uma contradição.
 
\end_layout

\begin_layout Section
Equivalência
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\begin_layout Standard
Aqui está uma noção similar útil: 
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status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset

 
\begin_inset Formula $\meta{A}$
\end_inset

 e 
\begin_inset Formula $\meta{B}$
\end_inset

 são 
\begin_inset ERT
status collapsed

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\backslash
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\end_inset


\begin_inset ERT
status collapsed

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{
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equivalentes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 (em LVF) sse, para qualquer valoração, os valores de verdade delas coincidem,
 ou seja, se não há naloração na qual elas tê valores de verdade opostos.
 
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
Na realidade, já usamos esta noção em 
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\backslash
S
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\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:MoreBracketingConventions"
plural "false"
caps "false"
noprefix "false"

\end_inset

; o ponto era que `
\begin_inset Formula $(A\eand B)\eand C$
\end_inset

' e `
\begin_inset Formula $A\eand(B\eand C)$
\end_inset

' são equivalentes.
 Novamente, é fácil fazer um teste para equivalência usando tabelas de verdade.
 Considere as sentenças `
\begin_inset Formula $\enot(P\eor Q)$
\end_inset

' e `
\begin_inset Formula $\enot P\eand\enot Q$
\end_inset

'.
 Elas são equivalentes? Para descobrir isso, construímos uma tabela de verdade.
 
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V 
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V
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Olhe para as colunas dos operadores lógicos principais; a negação para primeira
 sentença, conjunção para a segunda.
 Na primeiras três linhas, ambas sentenças são falsas.
 Na linha final, ambas são verdadeiras.
 Uma vez que elas coicidem em cada linha, as suas sentenças são equivalentes.
 
\end_layout

\begin_layout Section
Satisfatibilidade
\end_layout

\begin_layout Standard
Em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicNotions"
plural "false"
caps "false"
noprefix "false"

\end_inset

, disemos que sentenças são conjuntamente possíveis sse é possível que todas
 elas sejam verdadeiras ao mesmo tempo.
 Podemos oferecer um representante para esta noção também: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

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{
\end_layout

\end_inset

 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 são 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

conjuntamente satisfat
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'i
\end_layout

\end_inset

veis
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 (em LVF) sse há alguma valoração que as faz todas verdadeiras.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\end_layout

\begin_layout Standard
De forma derivada, sentenças são 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

conjuntamente insatisfa
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
' iveis
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 se não há nenhuma valoração que as faz todas verdadeiras.
 Novamente, é fácil fazer um teste para satisfatibilidade conjunta usando
 tabelas de verdade.
 
\end_layout

\begin_layout Section
Acarretamento e validade
\end_layout

\begin_layout Standard
A seguinte idea é intimamente relacionada àquela de satisfatibilidade conjunta:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 As sentenças 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

acarretam
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 (em LVF) a sentença 
\begin_inset Formula $\meta{C}$
\end_inset

 se não existir uma valoração das letras sentenciais que torne todas as
 sentenças 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 verdadeiras e 
\begin_inset Formula $\meta{C}$
\end_inset

 falsa.
 
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Novamente, é fácil fazer um teste para isto com uma tabela de verdade.
 Para checar se `
\begin_inset Formula $\enot L\eif(J\eor L)$
\end_inset

' e `
\begin_inset Formula $\enot L$
\end_inset

' acarretam `
\begin_inset Formula $J$
\end_inset

', basta simplesmente checar se há alguma valoração que faça ambos `
\begin_inset Formula $\enot L\eif(J\eor L)$
\end_inset

' e `
\begin_inset Formula $\enot L$
\end_inset

' verdadeiras, enquanto faça `
\begin_inset Formula $J$
\end_inset

' falsa.
 Desse modo, usamos a tabela de verdade:
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="11">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="f">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $J$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $L$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $L$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(J$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $L)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $L$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $J$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
A única linha na qual ambas `
\begin_inset Formula $\enot L\eif(J\eor L)$
\end_inset

' e `
\begin_inset Formula $\enot L$
\end_inset

' são verdadeiras é a segunda linha.
 Mas, nesta linha, `
\begin_inset Formula $J$
\end_inset

' também é verdadeira.
 Assim, `
\begin_inset Formula $\enot L\eif(J\eor L)$
\end_inset

' e `
\begin_inset Formula $\enot L$
\end_inset

' acarretam `
\begin_inset Formula $J$
\end_inset

'.
 
\end_layout

\begin_layout Standard
Fazemos agora uma observação importante: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 Se 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 acarretam 
\begin_inset Formula $\meta{C}$
\end_inset

 em LVF, então 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}\therefore\meta{C}$
\end_inset

 é válido.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Aqui está o motivo.
 Se 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 acarretam 
\begin_inset Formula $\meta{C}$
\end_inset

, então não há valoração que faça todas [sentenças] 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 verdadeiras e também faça 
\begin_inset Formula $\meta{C}$
\end_inset

 falsa.
 Qualquer caso no qual todas [as sentenças] 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 são verdadeiras e 
\begin_inset Formula $\meta{C}$
\end_inset

 é falsa geraria uma valoração com esta propriedade: tome o valor de verdade
 de qualquer letra sentencial como sendo apenas o valor de verdade da sentença
 correspondente nesse caso.
 Uma vez que não há tal valoração, não há caso no qual todas [sentenças]
 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 são verdadeiras e 
\begin_inset Formula $\meta{C}$
\end_inset

 é falsa.
 Mas isto é justamente o que faz com que um argumento com premissas 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 e conclusão 
\begin_inset Formula $\meta{C}$
\end_inset

 seja válido.
\end_layout

\begin_layout Standard
Resumindo, temos uma maneira de testar a validade de argumentos do Português.
 Em primeiro lugar, os argumentos são simbolizados em LVF, tendo premissas
 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 e conclusão 
\begin_inset Formula $\meta{C}$
\end_inset

.
 Então testamos acarretamento em LVF, usando tabelas de verdade.
\end_layout

\begin_layout Section
Os limites destes teste
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:ParadoxesOfMaterialConditional"

\end_inset

 Atingimos um marco importante: um teste para validade de argumentos! Contudo,
 não deveríamos ficar tão entusiasmados ainda.
 É importante entender os 
\emph on
limites
\emph default
 de nossa conquista.
 Ilustraremos estes limites com três exemplos.
\end_layout

\begin_layout Standard
Em primeiro lugar, considere o argumento: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Daisy tem quatro pernas.
 Portanto, Daisy tem mais de duas pernas.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Para simbolizar este argumento em LVF, teríamos de usar duas letras sentenciais
 diferentes — talvez `
\begin_inset Formula $F$
\end_inset

' e `
\begin_inset Formula $T$
\end_inset

' — para premissa e conclusão, respectivamente.
 Agora, é óbvio que `
\begin_inset Formula $F$
\end_inset

' não acarreta `
\begin_inset Formula $T$
\end_inset

'.
 O argumento do Português parece seguramente válido, não obstante! 
\end_layout

\begin_layout Standard
Em segundo lugar, considere a sentença:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
setcounter{eargnum}{1}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "n:JanBald"

\end_inset

 Jan é nem calvo nem não-calvo.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Para simbolizar esta sentença em LVF, ofereceríamos algo como `
\begin_inset Formula $\enot J\eand\enot\enot J$
\end_inset

'.
 Isto é uma contradição (cheque isto com uma tabela de verdade), mas a sentença
 
\begin_inset CommandInset ref
LatexCommand ref
reference "n:JanBald"
plural "false"
caps "false"
noprefix "false"

\end_inset

 não se parece com uma contradição; pois, poderíamos alegramente ter adicionado
 `Jan está na fronteira da calvície'! 
\end_layout

\begin_layout Standard
Em terceiro lugar, considere a seguinte sentença:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
setcounter{eargnum}{2}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "n:GodParadox"

\end_inset

 Não é o caso que se Deus existe, ele responde a orações malévolas.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Simbolizando isto em LVF, ofereceríamos algo como `
\begin_inset Formula $\enot(G\eif M)$
\end_inset

'.
 Ora, `
\begin_inset Formula $\enot(G\eif M)$
\end_inset

' acarreta `
\begin_inset Formula $G$
\end_inset

' (novamente, cheque isto com tabela de verdade).
 Assim, se simbolizarmos a sentença 
\begin_inset CommandInset ref
LatexCommand ref
reference "n:GodParadox"
plural "false"
caps "false"
noprefix "false"

\end_inset

 em LVF, ela parece implicar que Deus existe.
 Mas isto é estranho: certamente, até mesmo um ateu pode aceitar a sentença
 
\begin_inset CommandInset ref
LatexCommand ref
reference "n:GodParadox"
plural "false"
caps "false"
noprefix "false"

\end_inset

 sem se contradizer.
\end_layout

\begin_layout Standard
Uma lição disto é que a simbolização de 
\begin_inset CommandInset ref
LatexCommand ref
reference "n:GodParadox"
plural "false"
caps "false"
noprefix "false"

\end_inset

 como `
\begin_inset Formula $\enot(G\eif M)$
\end_inset

' mostra que V não expressa o que temos em mente.
 Talvez deveríamos parafraseá-la como 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
setcounter{eargnum}{2}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "n:GodParadox2"

\end_inset

 Se Deus existe, ele não responde a orações malévolas
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 e simbolizar 
\begin_inset CommandInset ref
LatexCommand ref
reference "n:GodParadox2"
plural "false"
caps "false"
noprefix "false"

\end_inset

 como `
\begin_inset Formula $G\eif\enot M$
\end_inset

'.
 Agora, se os ateus estiverem corretos e não há Deus, então `
\begin_inset Formula $G$
\end_inset

' é falsa e, assim, `
\begin_inset Formula $G\eif\enot M$
\end_inset

' é verdadeira e o quebra-cabeça desaparece.
 Todavia, se `
\begin_inset Formula $G$
\end_inset

' é falsa, então `
\begin_inset Formula $G\eif M$
\end_inset

', ou seja, `Se Deus existe, ele responde a orações malévolas' é 
\emph on
também
\emph default
 verdadeira! De formas distintas, estes três exemplos realçam alguns dos
 limites de se trabalhar com uma linguagem (como LVF) que pode 
\emph on
somente
\emph default
 lidar com conectivos verofuncionais.
 Além disso, estes limites dão origem a algumas questões interessantes em
 lógica filosófica.
 O caso da calvície de Jan (ou não) levanta a questão geral de qual lógica
 deveríamos usar quando lidamos com discurso 
\emph on
vago
\emph default
.
 O caso do ateu levanta a questão de como lidar com (os então chamados)
 
\emph on
paradoxos do condicional material
\emph default
.
 Parte do propósito deste curso é lhe dar as ferramentas necessárias para
 explorar estas questões da 
\emph on
lógica filosófica
\emph default
.
 Mas temos que andar, antes de podermos correr.
 Temos de nos tornar proficientes no uso de LVF, antes de podermos discutir
 adequadamente seus limites e considerar alternativas.
\end_layout

\begin_layout Section
A dupla catraca
\end_layout

\begin_layout Standard
Iremos usar a noção de acarretamento muitas vezes neste livro.
 Ajudar-nos-á, então, introduzir um símbolo que a abrevie.
 Em vez de dizer que as sentenças de LVF 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots$
\end_inset

 and 
\begin_inset Formula $\meta{A}_{n}$
\end_inset

 juntas acarretam 
\begin_inset Formula $\meta{C}$
\end_inset

, abreviaremos isto por:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}\entails\meta{C}
\]

\end_inset

O símbolo `
\begin_inset Formula $\entails$
\end_inset

' é conhecido por 
\emph on
dupla catraca
\emph default
, uma vez que se parece com uma catraca com duas barras horizontais.
 Vamos esclarecer o seguinte: `
\begin_inset Formula $\entails$
\end_inset

' não é um símbolo de LVF.
 Em vez disso, é um símbolo de nossa metalinguagem, Português estendido
 (lembre-se da diferença entre linguagem objeto e metalinguagem mencionada
 em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:UseMention"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
 Assim, a sentença metalinguística: 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{ebullet}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $P,P\eif Q\entails Q$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{ebullet}
\end_layout

\end_inset

 é somente uma abreviação para a sentença do Português:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{ebullet}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

As sentenças de LVF `
\begin_inset Formula $P$
\end_inset

' e `
\begin_inset Formula $P\eif Q$
\end_inset

' acarretam `
\begin_inset Formula $Q$
\end_inset

' 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{ebullet}
\end_layout

\end_inset

 Note que não há limite no números de sentenças de LVF que podem ser mencionadas
 antes do símbolo `
\begin_inset Formula $\entails$
\end_inset

'.
 De fato, podemos até mesmo considerar o caso limite: 
\begin_inset Formula 
\[
\entails\meta{C}
\]

\end_inset

Isto diz que não há valoração que faça todas as sentenças mencionadas à
 esquerda de `
\begin_inset Formula $\entails$
\end_inset

' verdadeiras, enquanto faça 
\begin_inset Formula $\meta{C}$
\end_inset

 falsa.
 Uma vez que nenhuma sentença é mencionada à esquerda de `
\begin_inset Formula $\entails$
\end_inset

' neste caso, isto significa justamente que não há valoração que faça 
\begin_inset Formula $\meta{C}$
\end_inset

 falsa.
 Dizendo de outra forma, é dito que qualquer valoração faz 
\begin_inset Formula $\meta{C}$
\end_inset

 verdadeiro.
 Ainda falando de outra forma, significa que 
\begin_inset Formula $\meta{C}$
\end_inset

 é uma tautologia.
 Similarmente:
\end_layout

\begin_layout Standard
Otherwise put, it says that every valuation makes 
\begin_inset Formula $\meta{C}$
\end_inset

 true.
 Otherwise put, it says that 
\begin_inset Formula $\meta{C}$
\end_inset

 is a tautology.
 Equally: 
\begin_inset Formula 
\[
\meta{A}\entails
\]

\end_inset

diz que 
\begin_inset Formula $\meta{A}$
\end_inset

 é uma contradição.
\end_layout

\begin_layout Section
`
\begin_inset Formula $\entails$
\end_inset

' versus `
\begin_inset Formula $\eif$
\end_inset

'
\end_layout

\begin_layout Standard
Queremos comparar e constratar `
\begin_inset Formula $\entails$
\end_inset

' e `
\begin_inset Formula $\eif$
\end_inset

'.
 
\end_layout

\begin_layout Standard
Observação: 
\begin_inset Formula $\meta{A}\entails\meta{C}$
\end_inset

 sse não há valoração das letras sentenciais que faça 
\begin_inset Formula $\meta{A}$
\end_inset

 verdadeira e 
\begin_inset Formula $\meta{C}$
\end_inset

 falsa.
 
\end_layout

\begin_layout Standard
Observação: 
\begin_inset Formula $\meta{A}\eif\meta{C}$
\end_inset

 é uma tautologia sse não existe valoração das letras sentenciais que faça
 
\begin_inset Formula $\meta{A}\eif\meta{C}$
\end_inset

 falsa.
 Uma vez que o condicional é verdadeiro exceto quando seu antecedente é
 verdaeiro e seu consequente, falso, 
\begin_inset Formula $\meta{A}\eif\meta{C}$
\end_inset

 é uma tautologia sse não há valoração que faça 
\begin_inset Formula $\meta{A}$
\end_inset

 verdadeira e 
\begin_inset Formula $\meta{C}$
\end_inset

 falsa.
 Combinando estas duas observações, vemos que 
\begin_inset Formula $\meta{A}\eif\meta{C}$
\end_inset

 é uma tautologia sse 
\begin_inset Formula $\meta{A}\entails\meta{C}$
\end_inset

.
 Mas há, de fato, uma importante diferença entre `
\begin_inset Formula $\entails$
\end_inset

' e `
\begin_inset Formula $\eif$
\end_inset

': 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Formula $\eif$
\end_inset

' é um conectivo sentencial de LVF.
\begin_inset Newline newline
\end_inset

 `
\begin_inset Formula $\entails$
\end_inset

' é um símbolo do Português estendido.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 De fato, quando `
\begin_inset Formula $\eif$
\end_inset

' é colocado entre duas sentenças de LVF, o resultado é uma sentença de
 LVF mais longa.
 Por outro lado, quando usamos `
\begin_inset Formula $\entails$
\end_inset

', formamos uma sentença metalinguística que 
\emph on
menciona
\emph default
 sentenças de LVF.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
practiceproblems
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Volte para suas respostas aos exercícios em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:CompleteTruthTables"
plural "false"
caps "false"
noprefix "false"

\end_inset


\series bold
A
\series default
.
 Determine quais eram tautologias, quais eram contradições e quais eram
 nem tautologias nem contradições.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
solutions
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.satisfiable"

\end_inset

 Use tabelas de verdade para determinar se estas sentenças são conjuntamente
 satisfatíveis ou são conjuntamente insatisfatíveis:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif A$
\end_inset

, 
\begin_inset Formula $\enot A\eif\enot A$
\end_inset

, 
\begin_inset Formula $A\eand A$
\end_inset

, 
\begin_inset Formula $A\eor A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $A\eif C$
\end_inset

, 
\begin_inset Formula $B\eif C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $B\eand(C\eor A)$
\end_inset

, 
\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $\enot(B\eor C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff(B\eor C)$
\end_inset

, 
\begin_inset Formula $C\eif\enot A$
\end_inset

, 
\begin_inset Formula $A\eif\enot B$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
solutions
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.valid"

\end_inset

 Use tabelas de verdade para determinar se cada argumento é válido ou inválido.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif A\therefore A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif(A\eand\enot A)\therefore\enot A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor(B\eif A)\therefore\enot A\eif\enot B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B,B\eor C,\enot A\therefore B\eand C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(B\eand A)\eif C,(C\eand A)\eif B\therefore(C\eand B)\eif A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Determine se cada sentença é uma tautologia, uma contradição ou uma sentença
 contingente, usando tabela de verdade completa.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot B\eand B$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot D\eor D$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eand B)\eor(B\eand A)$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot[A\eif(B\eif A)]$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff[A\eif(B\eand\enot B)]$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $[(A\eand B)\eiff B]\eif(A\eif B)$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.equiv"

\end_inset

 Determine se cada uma das sentenças seguintes são logicamente equivalentes,
 usando tabela de verdade completas.
 Se duas sentenças forem logicamente 
\begin_inset Quotes fls
\end_inset

equivalentes
\begin_inset Quotes frs
\end_inset

, escreva equivalente.
 Caso contrário, escreva 
\begin_inset Quotes fls
\end_inset

não equivalentes
\begin_inset Quotes frs
\end_inset

 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A$
\end_inset

 and 
\begin_inset Formula $\enot A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eand\enot A$
\end_inset

 and 
\begin_inset Formula $\enot B\eiff B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $[(A\eor B)\eor C]$
\end_inset

 and 
\begin_inset Formula $[A\eor(B\eor C)]$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor(B\eand C)$
\end_inset

 and 
\begin_inset Formula $(A\eor B)\eand(A\eor C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $[A\eand(A\eor B)]\eif B$
\end_inset

 and 
\begin_inset Formula $A\eif B$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.equiv2"

\end_inset

 Determine se cada uma das sentenças seguintes são logicamente equivalentes,
 usando tabela de verdade completas.
 Se duas sentenças forem logicamente 
\begin_inset Quotes fls
\end_inset

equivalentes
\begin_inset Quotes frs
\end_inset

, escreva equivalente.
 Caso contrário, escreva 
\begin_inset Quotes fls
\end_inset

não equivalentes
\begin_inset Quotes frs
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif A$
\end_inset

 and 
\begin_inset Formula $A\eiff A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eif B)$
\end_inset

 and 
\begin_inset Formula $\enot A\eif\enot B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

 and 
\begin_inset Formula $\enot A\eif B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eif B)\eif C$
\end_inset

 and 
\begin_inset Formula $A\eif(B\eif C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff(B\eiff C)$
\end_inset

 and 
\begin_inset Formula $A\eand(B\eand C)$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.satisfiable2"

\end_inset

 Determine se cada coleção de sentenças é conjuntamente satisfatíveis ou
 conjuntamente insatisfatíveis, usando tabela de verdade completa.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eand\enot B$
\end_inset

, 
\begin_inset Formula $\enot(A\eif B)$
\end_inset

, 
\begin_inset Formula $B\eif A$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $A\eif\enot A$
\end_inset

, 
\begin_inset Formula $B\eif\enot B$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(\enot A\eor B)$
\end_inset

, 
\begin_inset Formula $A\eif\enot C$
\end_inset

, 
\begin_inset Formula $A\eif(B\eif C)$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $A\eand\enot B$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif(B\eif C)$
\end_inset

, 
\begin_inset Formula $(A\eif B)\eif C$
\end_inset

, 
\begin_inset Formula $A\eif C$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.satisfiable3"

\end_inset

 Determine se cada coleção de sentenças é conjuntamente satisfatíveis ou
 conjuntamente insatisfatíveis, usando tabela de verdade completa.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot B$
\end_inset

, 
\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $A$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eor B)$
\end_inset

, 
\begin_inset Formula $A\eiff B$
\end_inset

, 
\begin_inset Formula $B\eif A$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $\enot B$
\end_inset

, 
\begin_inset Formula $\enot B\eif\enot A$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff B$
\end_inset

, 
\begin_inset Formula $\enot B\eor\enot A$
\end_inset

, 
\begin_inset Formula $A\eif B$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eor B)\eor C$
\end_inset

, 
\begin_inset Formula $\enot A\eor\enot B$
\end_inset

, 
\begin_inset Formula $\enot C\eor\enot B$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.valid2"

\end_inset

 Determine se cada argumento é válido ou inválido, usando tabela de verdade
 completa.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $B\therefore A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff B$
\end_inset

, 
\begin_inset Formula $B\eiff C\therefore A\eiff C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $A\eif C\therefore B\eif C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $B\eif A\therefore A\eiff B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.valid3"

\end_inset

 Determine se cada argumento é válido ou inválido, usando tabela de verdade
 completa.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor\bigl[A\eif(A\eiff A)\bigr]\therefore A$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $B\eor C$
\end_inset

, 
\begin_inset Formula $\enot B\therefore A\eand C$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $\enot A\therefore\enot B$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A$
\end_inset

, 
\begin_inset Formula $B\therefore\enot(A\eif\enot B)$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eand B)$
\end_inset

, 
\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $A\eiff B\therefore C$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
solutions
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.concepts"

\end_inset

 Responda cada uma das questões abaixo e justifique sua resposta.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Suponha que 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 são logicamente equivalentes.
 O que podemos dizer sobre 
\begin_inset Formula $\meta{A}\eiff\meta{B}$
\end_inset

?
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Suponha que 
\begin_inset Formula $(\meta{A}\eand\meta{B})\eif\meta{C}$
\end_inset

 é nem tautologia nem contradição.
 
\begin_inset Formula $\meta{A},\meta{B}\therefore\meta{C}$
\end_inset

 é válido?
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Suponha que 
\begin_inset Formula $\meta{A}$
\end_inset

, 
\begin_inset Formula $\meta{B}$
\end_inset

 e 
\begin_inset Formula $\meta{C}$
\end_inset

 são conjuntamente insatisfatíveis.
 O que podemos dizer sobre 
\begin_inset Formula $(\meta{A}\eand\meta{B}\eand\meta{C})$
\end_inset

?
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Suponha que 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 é uma contradição.
 O que podemos dizer sobre se 
\begin_inset Formula $\meta{A},\meta{B}\entails\meta{C}$
\end_inset

?
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Suponha que 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 é uma tautologia.
 O que podemos dizer sobre se 
\begin_inset Formula $\meta{A},\meta{B}\entails\meta{C}$
\end_inset

?
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Suponha que 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 são logicamente equivalentes.
 O que podemos dizer sobre 
\begin_inset Formula $(\meta{A}\eor\meta{B})$
\end_inset

?
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Suponha que 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\emph on
não
\emph default
 são logicamente equivalentes.
 O que podemos dizer sobre 
\begin_inset Formula $(\meta{A}\eor\meta{B})$
\end_inset

?
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Considere o seguinte princípio:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{ebullet}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Suponha que 
\begin_inset Formula $\meta{A}$
\end_inset

 e 
\begin_inset Formula $\meta{B}$
\end_inset

 são logicamente equivalentes.
 Suponha que um argumento contém 
\begin_inset Formula $\meta{A}$
\end_inset

 (ou como premissa ou como conclusão).
 A validade do argumento não seria afetada, se substituíssemos 
\begin_inset Formula $\meta{A}$
\end_inset

 por 
\begin_inset Formula $\meta{B}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{ebullet}
\end_layout

\end_inset

 Este princípio é correto? Explique sua resposta.
\end_layout

\begin_layout Chapter
Atalhos na tabela de verdade
\end_layout

\begin_layout Standard
Com a prática, rapidamente você torna-se-á apto a preencher as tabelas de
 verdade.
 Neste capítulo, queremos apresentar atalhos permitidos que o ajudrão ao
 longo do percurso.
 
\end_layout

\begin_layout Section
Trabalhando com tabelas de verdade
\end_layout

\begin_layout Standard
Vovê descobrirá rapidamente que você não precisa copiar os valores de verdade
 de cada sentença, mas pode simplesmente se referir a eles [
\emph on
but can simply refer back to them.
\emph default
].
 Assim, podemos acelerar as coisas escrevendo: 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="8">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $P$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $Q$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(P$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $Q)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eiff
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $P$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

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\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

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Similarmente, com certeza, sabemos que uma conjunção é falsa, sempre que
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\end_layout

\begin_layout Standard
Um atalho similar está disponível para condicionais.
 Você sabe imediatamente que um condicional é verdadeiro se o consequente
 é verdadeiro ou o antecedente é falso.
 Desse modo, poderíamos apresentar: 
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F 
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F 
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F 
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V
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\end_layout

\begin_layout Standard
Logo, `
\begin_inset Formula $((P\eif Q)\eif P)\eif P$
\end_inset

' é uma tautologia.
 De fato, é uma instância da 
\emph on
Lei de Peirce
\emph default
, nomeada assim por causa de Charles Sanders Peirce.
 
\end_layout

\begin_layout Section
Teste para validade e acarretamento
\end_layout

\begin_layout Standard
Quando usamos tabelas de verdade para testar validade e acarretamento, estamos
 checando as linhas 
\emph on
ruins
\emph default
: as linhas onde as premissas são todas verdadeiras e a conclusão falsa.
 Note:
\end_layout

\begin_layout Standard
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status collapsed

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\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
item[$
\backslash
bullet$]
\end_layout

\end_inset

Qualquer linha em que a conclusão é verdadeira não é uma linha ruim.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
item[$
\backslash
bullet$]
\end_layout

\end_inset

Qualquer linha em que alguma premissa é falsa não é uma linha ruim.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Uma vez que 
\emph on
tudo
\emph default
 que estamos fazendo é procurar pelas linhas ruins, deveríamos ter em mente
 o seguinte: se você encontra uma linha ode a conclusão é verdadeira, não
 precisamos avaliar qualquer outra coisa nesta linha.
 Esta linha não é definitivamente ruim.
 Da mesma forma, se encontramos um linha onde alguma premissa é falsa, não
 precisamos avaliar qualquer outra coisa nesta linha.
 
\end_layout

\begin_layout Standard
Com isto em mente, considere agora como poderíamos testar a validade do
 sehuinte argumento: 
\begin_inset Formula 
\[
\enot L\eif(J\eor L),\enot L\therefore J
\]

\end_inset

A 
\emph on
primeira
\emph default
 coisa que deveríamos fazer é avaliar a conclusão.
 Se encontramos que a conclusão é 
\emph on
verdadeira
\emph default
 em alguma linha, então esta linha não é ruim.
 Desse modo, podemos simplesmente ignorar o restante da linha.
 Assim, em nosso primeiro estágio, temos algo parecido com: 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
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<column alignment="none" valignment="top" special="f">
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V 
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\begin_layout Standard
A tabela de verdade não tem linhas ruins, portanto o argumento é válido
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\begin_layout Standard
Vale a pena ilustrar a tática novamente.
 Vamos checar se o seguinte argumento é válido 
\begin_inset Formula 
\[
A\eor B,\enot(A\eand C),\enot(B\eand\enot D)\therefore(\enot C\eor D)
\]

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Agora devemos avaliar as premissas.
 Usamos os atalhos onde é possível:
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\emph on
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Tabelas de verdade parciais
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Às vezes, não precisamos saber o que acontece em qualquer linha da tabela
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Tautologia.
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tabela de verdade parcial
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<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
Agora, precisamos apenas tornar `
\begin_inset Formula $(S\eand W)$
\end_inset

' falsa.
 Para fazer isto, precisamos tornar pelo menos uma das sentenças `
\begin_inset Formula $S$
\end_inset

' e `
\begin_inset Formula $W$
\end_inset

' falsa.
 Podemos fazer tanto `
\begin_inset Formula $S$
\end_inset

' como `
\begin_inset Formula $W$
\end_inset

' falsas, se quisermos.
 Tudo que importa é que a sentença inteira se torne falsa nesta linha.
 Tomando-se uma decisão arbitrária, teminamos a tabelas da seguinte forma:
 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="11">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $S$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $U$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $W$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(U$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(S$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $W)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
T 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
T 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
T 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
T 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
T 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
Agora temos uma tabela de verdade parcial que mostra que `
\begin_inset Formula $(U\eand T)\eif(S\eand W)$
\end_inset

' não é uma tautologia.
 Falando de outra forma, mostramos que existe uma valoração qie faz `
\begin_inset Formula $(U\eand T)\eif(S\eand W)$
\end_inset

' falsa, a saber, a valoração que faz `
\begin_inset Formula $S$
\end_inset

' falsa, `
\begin_inset Formula $T$
\end_inset

' verdadeira, `
\begin_inset Formula $U$
\end_inset

' verdadeira e `
\begin_inset Formula $W$
\end_inset

' falsa.
\end_layout

\begin_layout Paragraph
Contradição.
\end_layout

\begin_layout Standard
Mostrar que algo é uma contradição exige uma tabela completa: precisamos
 mostrar que não há valoração que faça a sentença verdadeira; ou seja, precisamo
s mostrar que a sentença é falsa em qualquer linha da tabela de verdade.
 Entretanto, para mostrar que algo 
\emph on
não
\emph default
 é uma contradição, tudo que precisamos fazer é encontrar uma valoração
 que faça a sentença verdadeira e uma única linha de uma tabela de verdade
 será suficiente.
 Podemos ilustrar isto com o mesmo exemplo: 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="11">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $S$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $U$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $W$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(U$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(S$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $W)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
Para tornar a sentença verdadeira, será suficiente garantir que o antecedente
 é falso.
 Uma vez que o antecedente é uma conjunção, podemos tornar um dos conjunctos
 falso.
 Arbitrariamente, escolhemos tornar `
\begin_inset Formula $U$
\end_inset

' falsa; e, então, podemos atribuir qualquer valor de verdade que quisermos
 às outrasletras sentenciais.
 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="11">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="none" valignment="top" special="d">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="e">
<column alignment="none" valignment="top" special="f">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $S$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $U$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $W$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(U$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(S$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $W)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
V 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
TTbf
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

V
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Paragraph
Equivalência.
\end_layout

\begin_layout Standard
Para mostrar que duas sentenças são equivalentes, devemos mostrar que as
 sentenças tê o mesmo valor de verdade em qualquer valoração.
 Isto exige uma tabela de verdade completa.
 
\end_layout

\begin_layout Standard
Para mostrar que duas sentenças 
\emph on
não
\emph default
 são equivalentes, apenas precisamos mostrar que existe uma valoração na
 qual elas têm diferentes valores de verdade.
 Assim, isto requer apenas uma tabela de verdade parcial de uma linha: faça
 uma tabela de forma que uma sentença seja verdadeira e a outra, falsa.
\end_layout

\begin_layout Paragraph
Consistência.
\end_layout

\begin_layout Standard
Para mostrar que algumas sentenças são conjuntamente satisfatíveis, devemos
 mostrar que há uma valoração que torne todas as sentenças verdadeiras,
 portanto isto exige apenas uma tabela de verdade parcial com uma única
 linha.
 
\end_layout

\begin_layout Standard
Para mostrar que algumas sentenças são conjuntamente insatisfatíveis, devemos
 mostrar que não há valoração que faça todas as sentenças verdadeiras.
 Isto requer uma tabela de verdade completa: você deve mostrar que, em toda
 linha da tabela de verdade, pelo menos uma das sentenças é falsa.
\end_layout

\begin_layout Paragraph
Validade.
\end_layout

\begin_layout Standard
Para mostrar que um argumento é válido, devemos mostrar que não há valoração
 que faça todas as premissas verdadeiras e a conclusão falsa.
 Assim, isto exige uma tabela de verdade completa (similarmente para acarretamen
to).
 Para mostrar que um argumento é 
\emph on
inválido
\emph default
, devemos mostrar que há uma valoração que faça todas as premissas verdadeiras
 e a conclusão falsa.
 Desse modo, isto requer apenas uma tabela de verdade parcial com uma única
 linha na qual todas as premissas são verdadeiras e a conclusão é falsa
 (similarmente para acarretamento).
\end_layout

\begin_layout Standard
Esta tabela resume o que é requerido:
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="3">
<features tabularvalignment="middle">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\series bold
Sim
\series default
 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\series bold
Não
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
tautologia? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
completa 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
parcial com uma linha 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
contradição? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
completa 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
parcial com uma linha 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
equivalente? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
complete 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
parcial com uma linha 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
satisfatível? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
parcial com uma linha 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
completa 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
válido? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
completa 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
parcial com uma linha 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
acarretamento? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
completa 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
parcial com uma linha 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "table.CompleteVsPartial"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
practiceproblems
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
solutions
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
solutions
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.equiv3"

\end_inset

 Use tabelas de verdade parciais ou completas (conforme apropriado) para
 determinar se estes pares de sentenças são logicamente equivalentes: 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A$
\end_inset

, 
\begin_inset Formula $\enot A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A$
\end_inset

, 
\begin_inset Formula $A\eor A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif A$
\end_inset

, 
\begin_inset Formula $A\eiff A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor\enot B$
\end_inset

, 
\begin_inset Formula $A\eif B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eand\enot A$
\end_inset

, 
\begin_inset Formula $\enot B\eiff B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eand B)$
\end_inset

, 
\begin_inset Formula $\enot A\eor\enot B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eif B)$
\end_inset

, 
\begin_inset Formula $\enot A\eif\enot B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eif B)$
\end_inset

, 
\begin_inset Formula $(\enot B\eif\enot A)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
solutions
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.satisfiable4"

\end_inset

 Use tabelas de verdade parciais ou completas (conforme apropriado) para
 determinar estas sentenças são conjuntamente satisfatíveis ou conjuntamente
 insatisfatíveis: 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eand B$
\end_inset

, 
\begin_inset Formula $C\eif\enot B$
\end_inset

, 
\begin_inset Formula $C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $B\eif C$
\end_inset

, 
\begin_inset Formula $A$
\end_inset

, 
\begin_inset Formula $\enot C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $B\eor C$
\end_inset

, 
\begin_inset Formula $C\eif\enot A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A$
\end_inset

, 
\begin_inset Formula $B$
\end_inset

, 
\begin_inset Formula $C$
\end_inset

, 
\begin_inset Formula $\enot D$
\end_inset

, 
\begin_inset Formula $\enot E$
\end_inset

, 
\begin_inset Formula $F$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eand(B\eor C)$
\end_inset

, 
\begin_inset Formula $\enot(A\eand C)$
\end_inset

, 
\begin_inset Formula $\enot(B\eand C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $B\eif C$
\end_inset

, 
\begin_inset Formula $\enot(A\eif C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
solutions
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.valid4"

\end_inset

 Use tabelas de verdade parciais ou completas (conforme apropriado) para
 determinar se cada argumento é válido ou inválido: 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor\bigl[A\eif(A\eiff A)\bigr]\therefore A$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff\enot(B\eiff A)\therefore A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B,B\therefore A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B,B\eor C,\enot B\therefore A\eand C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff B,B\eiff C\therefore A\eiff C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.TTorC3"

\end_inset

 Determine se cada sentença é uma tautologia, uma contradição ou uma contingênci
a.
 Justifique sua resposta com uma tabela de verdade completa ou parcial conforme
 apropriado.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif\enot A$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif(A\eand(A\eor B))$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eif B)\eiff(B\eif A)$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif\enot(A\eand(A\eor B))$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot B\eif[(\enot A\eand A)\eor B]$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eor B)\eiff(\enot A\eand\enot B)$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $[(A\eand B)\eand C]\eif B$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot\bigl[(C\eor A)\eor B\bigr]$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\bigl[(A\eand B)\eand\enot(A\eand B)\bigr]\eand C$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eand B)]\eif[(A\eand C)\eor(B\eand D)]$
\end_inset

 
\begin_inset VSpace 0.5ex
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.TTorC4"

\end_inset

 Determine se cada sentença é uma tautologia, uma contradição ou uma contingênci
a.
 Justifique sua resposta com uma tabela de verdade completa ou parcial conforme
 apropriado.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eor A)$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eif B)\eor(B\eif A)$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $[(A\eif B)\eif A]\eif A$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot[(A\eif B)\eor(B\eif A)]$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eand B)\eor(A\eor B)$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eand B)\eiff A$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif(B\eor C)$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eand\enot A)\eif(B\eor C)$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(B\eand D)\eiff[A\eiff(A\eor C)]$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot[(A\eif B)\eor(C\eif D)]$
\end_inset


\begin_inset VSpace 0.5ex
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Determine se os seguintes pares de sentenças são logicamente equivalentes,
 usando tabelas de verdade completas.
 Se as duas sentenças forem, de fato, equivalentes, escreva 
\begin_inset Quotes fls
\end_inset

equivalente
\begin_inset Quotes frs
\end_inset

.
 Caso contrário, escreva 
\begin_inset Quotes fls
\end_inset

não equivalente
\begin_inset Quotes frs
\end_inset

.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A$
\end_inset

 and 
\begin_inset Formula $A\eor A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A$
\end_inset

 and 
\begin_inset Formula $A\eand A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor\enot B$
\end_inset

 and 
\begin_inset Formula $A\eif B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eif B)$
\end_inset

 and 
\begin_inset Formula $(\enot B\eif\enot A)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eand B)$
\end_inset

 and 
\begin_inset Formula $\enot A\eor\enot B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $((U\eif(X\eor X))\eor U)$
\end_inset

 and 
\begin_inset Formula $\enot(X\eand(X\eand U))$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $((C\eand(N\eiff C))\eiff C)$
\end_inset

 and 
\begin_inset Formula $(\enot\enot\enot N\eif C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $[(A\eor B)\eand C]$
\end_inset

 and 
\begin_inset Formula $[A\eor(B\eand C)]$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $((L\eand C)\eand I)$
\end_inset

 and 
\begin_inset Formula $L\eor C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.satisfiable5"

\end_inset

 Determine se cada coleção de sentenças é conjntamente satisfatível ou conjuntam
ente insatisfatível.
 Justifique sua resposta com uma tabela de verdade completa ou parcial conforme
 apropriado.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif A$
\end_inset

, 
\begin_inset Formula $\enot A\eif\enot A$
\end_inset

, 
\begin_inset Formula $A\eand A$
\end_inset

, 
\begin_inset Formula $A\eor A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif\enot A$
\end_inset

, 
\begin_inset Formula $\enot A\eif A$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $A\eif C$
\end_inset

, 
\begin_inset Formula $B\eif C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $A\eif C$
\end_inset

, 
\begin_inset Formula $B\eif C$
\end_inset

, 
\begin_inset Formula $\enot C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $B\eand(C\eor A)$
\end_inset

, 
\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $\enot(B\eor C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eiff B)\eif B$
\end_inset

, 
\begin_inset Formula $B\eif\enot(A\eiff B)$
\end_inset

, 
\begin_inset Formula $A\eor B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff(B\eor C)$
\end_inset

, 
\begin_inset Formula $C\eif\enot A$
\end_inset

, 
\begin_inset Formula $A\eif\enot B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff B$
\end_inset

, 
\begin_inset Formula $\enot B\eor\enot A$
\end_inset

, 
\begin_inset Formula $A\eif B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff B$
\end_inset

, 
\begin_inset Formula $A\eif C$
\end_inset

, 
\begin_inset Formula $B\eif D$
\end_inset

, 
\begin_inset Formula $\enot(C\eor D)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eand\enot B)$
\end_inset

, 
\begin_inset Formula $B\eif\enot A$
\end_inset

, 
\begin_inset Formula $\enot B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Determine se cada argumento é válido ou inválido.
 Justifique sua resposta com uma tabela de verdade completa ou parcial conforme
 apropriado.
 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.valid5"

\end_inset

 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif(A\eand\enot A)\therefore\enot A$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $B\eif A\therefore A\eiff B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor(B\eif A)\therefore\enot A\eif\enot B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $B\eif A\therefore A\eand B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(B\eand A)\eif C$
\end_inset

, 
\begin_inset Formula $(C\eand A)\eif B\therefore(C\eand B)\eif A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(\enot A\eor\enot B)$
\end_inset

, 
\begin_inset Formula $A\eif\enot C\therefore A\eif(B\eif C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eand(B\eif C)$
\end_inset

, 
\begin_inset Formula $\enot C\eand(\enot B\eif\enot A)\therefore C\eand\enot C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eand B$
\end_inset

, 
\begin_inset Formula $\enot A\eif\enot C$
\end_inset

, 
\begin_inset Formula $B\eif\enot D\therefore A\eor B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B\therefore(A\eand B)\eor(\enot A\eand\enot B)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot A\eif B$
\end_inset

,
\begin_inset Formula $\enot B\eif C$
\end_inset

,
\begin_inset Formula $\enot C\eif A\therefore\enot A\eif(\enot B\eor\enot C)$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Determine se cada argumento é válido ou inválido.
 Justifique sua resposta com uma tabela de verdade completa ou parcial conforme
 apropriado.
 
\begin_inset CommandInset label
LatexCommand label
name "pr.TT.valid6"

\end_inset

 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff\enot(B\eiff A)\therefore A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $B\eor C$
\end_inset

, 
\begin_inset Formula $\enot A\therefore B\eand C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif C$
\end_inset

, 
\begin_inset Formula $E\eif(D\eor B)$
\end_inset

, 
\begin_inset Formula $B\eif\enot D\therefore(A\eor C)\eor(B\eif(E\eand D))$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B$
\end_inset

, 
\begin_inset Formula $C\eif A$
\end_inset

, 
\begin_inset Formula $C\eif B\therefore A\eif(B\eif C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B$
\end_inset

, 
\begin_inset Formula $\enot B\eor A\therefore A\eiff B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 
\end_layout

\end_body
\end_document
